Newform orbit 345.2.h.b

• Label: 345.2.h.b
• Level: $345$
• Weight: $2$
• Character orbit: 345.h
• Analytic conductor: $2.755$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 345 = 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	345.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.75483886973\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.3057647616.5
Defining polynomial:	\( x^{8} - 4x^{7} + 8x^{5} + 10x^{4} + 12x^{2} + 6 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{2} + \beta_{2} q^{3} + ( - \beta_{6} + 1) q^{4} + ( - \beta_{4} - \beta_{3} - \beta_1) q^{5} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{6} - 3 q^{7} + ( - \beta_{7} + \beta_{2}) q^{8} + (\beta_{6} + \beta_{4} + 2 \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 24 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{5} + 10x^{4} + 12x^{2} + 6 \) :

\(\beta_{1}\)	\(=\)	\( ( -81\nu^{7} + 725\nu^{6} - 1250\nu^{5} - 2413\nu^{4} + 3778\nu^{3} + 6007\nu^{2} + 2166\nu + 2419 ) / 3445 \)
\(\beta_{2}\)	\(=\)	\( ( -197\nu^{7} + 700\nu^{6} + 575\nu^{5} - 2211\nu^{4} - 4549\nu^{3} + 3424\nu^{2} + 3992\nu - 1347 ) / 3445 \)
\(\beta_{3}\)	\(=\)	\( ( 361\nu^{7} - 1615\nu^{6} + 765\nu^{5} + 2163\nu^{4} + 3492\nu^{3} + 703\nu^{2} + 554\nu - 4614 ) / 3445 \)
\(\beta_{4}\)	\(=\)	\( ( -368\nu^{7} + 1465\nu^{6} - 150\nu^{5} - 2329\nu^{4} - 3846\nu^{3} - 354\nu^{2} - 7512\nu - 68 ) / 3445 \)
\(\beta_{5}\)	\(=\)	\( ( -34\nu^{7} + 180\nu^{6} - 155\nu^{5} - 352\nu^{4} + 22\nu^{3} + 673\nu^{2} - 406\nu + 276 ) / 265 \)
\(\beta_{6}\)	\(=\)	\( ( 46\nu^{7} - 150\nu^{6} - 180\nu^{5} + 523\nu^{4} + 812\nu^{3} - 22\nu^{2} + 144\nu + 141 ) / 265 \)
\(\beta_{7}\)	\(=\)	\( ( 769\nu^{7} - 2715\nu^{6} - 1615\nu^{5} + 6917\nu^{4} + 9853\nu^{3} + 3492\nu^{2} + 6486\nu + 3999 ) / 3445 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/345\mathbb{Z}\right)^\times\).

\(n\)	\(116\)	\(166\)	\(277\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

344.1

−1.16225	+	0.707107i
−1.16225	−	0.707107i
2.90421	+	0.707107i
2.90421	−	0.707107i
−0.172164	−	0.707107i
−0.172164	+	0.707107i
0.430200	−	0.707107i
0.430200	+	0.707107i

−2.17533

−0.796225

−

1.53819i

2.73205

−2.17533

−

0.517638i

1.73205

+

3.34607i

−3.00000

−1.59245

−1.73205

+

2.44949i

4.73205

+

1.12603i

344.2

−2.17533

−0.796225

+

1.53819i

2.73205

−2.17533

+

0.517638i

1.73205

−

3.34607i

−3.00000

−1.59245

−1.73205

−

2.44949i

4.73205

−

1.12603i

344.3

−1.12603

1.53819

−

0.796225i

−0.732051

−1.12603

+

1.93185i

−1.73205

+

0.896575i

−3.00000

3.07638

1.73205

−

2.44949i

1.26795

−

2.17533i

344.4

−1.12603

1.53819

+

0.796225i

−0.732051

−1.12603

−

1.93185i

−1.73205

−

0.896575i

−3.00000

3.07638

1.73205

+

2.44949i

1.26795

+

2.17533i

344.5

1.12603

−1.53819

−

0.796225i

−0.732051

1.12603

−

1.93185i

−1.73205

−

0.896575i

−3.00000

−3.07638

1.73205

+

2.44949i

1.26795

−

2.17533i

344.6

1.12603

−1.53819

+

0.796225i

−0.732051

1.12603

+

1.93185i

−1.73205

+

0.896575i

−3.00000

−3.07638

1.73205

−

2.44949i

1.26795

+

2.17533i

344.7

2.17533

0.796225

−

1.53819i

2.73205

2.17533

+

0.517638i

1.73205

−

3.34607i

−3.00000

1.59245

−1.73205

−

2.44949i

4.73205

+

1.12603i

344.8

2.17533

0.796225

+

1.53819i

2.73205

2.17533

−

0.517638i

1.73205

+

3.34607i

−3.00000

1.59245

−1.73205

+

2.44949i

4.73205

−

1.12603i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
115.c	odd	2	1	inner
345.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	345.2.h.b	✓	8
3.b	odd	2	1	inner	345.2.h.b	✓	8
5.b	even	2	1		345.2.h.c	yes	8
15.d	odd	2	1		345.2.h.c	yes	8
23.b	odd	2	1		345.2.h.c	yes	8
69.c	even	2	1		345.2.h.c	yes	8
115.c	odd	2	1	inner	345.2.h.b	✓	8
345.h	even	2	1	inner	345.2.h.b	✓	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
345.2.h.b	✓	8	1.a	even	1	1	trivial
345.2.h.b	✓	8	3.b	odd	2	1	inner
345.2.h.b	✓	8	115.c	odd	2	1	inner
345.2.h.b	✓	8	345.h	even	2	1	inner
345.2.h.c	yes	8	5.b	even	2	1
345.2.h.c	yes	8	15.d	odd	2	1
345.2.h.c	yes	8	23.b	odd	2	1
345.2.h.c	yes	8	69.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\):

\( T_{2}^{4} - 6T_{2}^{2} + 6 \)

\( T_{7} + 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - 6 T^{2} + 6)^{2} \)
$3$	\( T^{8} + 6T^{4} + 81 \)
$5$	T^8 - 4*T^6 + 6*T^4 - 100*T^2 + 625
$7$	\( (T + 3)^{8} \)
$11$	\( (T^{4} - 18 T^{2} + 6)^{2} \)